Question 426300
I assume the function is:
{{{R(t) = 1150/(0.5+22.5(2.2)^(-0.065t))}}}<br>
To answer the question about the rabbit population approaching some number "as time goes on" it means, in mathematical terms, as t approaches infinity.<br>
To answer this question we use some logic and some knowledge about how exponents and fractions work:<ul><li>As t approaches infinity (i.e a very, very large number), the exponent, -0.065t, will be become a very, very large negative number.</li><li>Since negative exponents mean reciprocals, 2.2 to a very large negative power is a fraction: 1 over 2.2 to a very, very large positive exponent.</li><li>As t gets larger and larger, the denominator of this fraction gets larger and larger.</li><li>As the denominator of a fraction gets larger and larger (without a change in the numerator), the fraction gets smaller and smaller. In fact it gets closer and closer to zero.</li><li>So as t gets larger and larger, {{{2.2^(-0.065t)}}} gets closer and closer to zero.</li><li>As {{{2.2^(-0.065t)}}} gets closer to zero, {{{22.5*2.2^(-0.065t)}}} also gets closer and closer to zero.</li><li>As {{{22.5*2.2^(-0.065t)}}} gets closer and closer to zero, the denominator, {{{0.5 + 22.5*2.2^(-0.065t)}}}, gets closer and closer to 0.5.</li><li>As the denominator gets closer and closer to 0.5, R(t) gets closer and closer to {{{1150/0.5}}}</li><li>And since {{{1150/0.5 = 2300}}}, the population of rabbits will approach 2300 "as time goes by".</li></ul>
P.S. R(t) only approaches 2300 for large values of t. It will never actually be equal to 2300. When you got an answer of 2300 for the population after 5 years you must have rounded off your answer to get 2300 (or you made an error.)